Positive definiteness and the Stolarsky invariance principle

Dmitriy Bilyk, Ryan W. Matzke, Oleksandr Vlasiuk

Research output: Contribution to journalArticlepeer-review


In this paper we elaborate on the interplay between energy optimization, positive definiteness, and discrepancy. In particular, assuming the existence of a K-invariant measure μ with full support, we show that conditional positive definiteness of a kernel K is equivalent to a long list of other properties: including, among others, convexity of the energy functional, inequalities for mixed energies, and the fact that μ minimizes the energy integral in various senses. In addition, we prove a very general form of the Stolarsky Invariance Principle on compact spaces, which connects energy minimization and discrepancy and extends several previously known versions.

Original languageEnglish (US)
Article number126220
JournalJournal of Mathematical Analysis and Applications
Issue number2
StatePublished - Sep 15 2022

Bibliographical note

Funding Information:
The research of the first author (D. Bilyk) received partial funding from the National Science Foundation (grants DMS-1665007 and DMS-2054606 ) and the Simons Foundation (Collaboration Grant 712810 ). O. Vlasiuk has been supported by an AMS-Simons Travel grant for early career mathematicians. In addition, the authors were supported by ICERM ( Brown University ) and NSF for a research in groups meeting.

Publisher Copyright:
© 2022 Elsevier Inc.


  • Discrepancy
  • Minimal energy
  • Positive definiteness
  • Stolarsky principle


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